Wave Equation¶

1-D Wave Equation¶

Separate Variable Method¶

• 前提

wave equation is on a closed interval $[0,L]$

assume that

then

finally we have

zero initial velocity¶

let $f(x)$ be an initial displacement function

then we can get

and finally,

• notice that p is a half of period ($[-p,p]$)

in which $a_n$ is the sine Fourier series coefficient of $f(x)=y(x,0)=\sum a_n\sin (\frac{n\pi x}{p})$

zero initial displacement¶

let $g(x)$ be the initial velocity function

then we can get

in which $\frac{n\pi c}{p}{b}_n$ is the Fourier sine series coefficient of $g(x)$

finally,

nonzero initial velocity and displacement¶

in the situation, assume

which $y_1(x,t)$ assume zero initial velocity, and that $y_2(x,t)$ assume zero initial displacement

and that is same as what we have discussed above.

with External Force¶

consider wave equation in this form

$y(x,t)$ can not be separated.

so we let

assume that $Y(x,t)$ can be separated, which means we can get the answer with the method above.

代入 $y(x,t)$, we can get

therefore

Infinite Medium¶

for the wave equations which are define in $(-\mathrm{\infty },\mathrm{\infty })$

as usual, denote the initial displacement as $f(x)$ and the initial velocity as $g(x)$

first we have to know that the eigenvalue only exists when $\omega \ge 0$

• solve two ODE we can get

then the most important thing is that

Semi-infinite Medium¶

for the wave equations which are define in $[0,\mathrm{\infty })$, thus

as usual, denote the initial displacement as $f(x)$ and the initial velocity as $g(x)$

%% since $X(0)=0$, thus

%%

Finite V.S. Infinite Medium¶

Characteristics and d'Alembert's Solution¶

one special solution for these condition

背

Forced Wave Motion¶

• solution

$\mathrm{\Delta }$ is characteristics triangle

$k_1$, $k_2$ can be any real constants.

2-D Wave Equation¶